<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <!-- hexo-inject:begin --><!-- hexo-inject:end --><meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#49b1f5">
<meta name="generator" content="Hexo 5.3.0">
  <link rel="apple-touch-icon" sizes="180x180" href="/images/favicon-180x180-yaowan.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-yaowan.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-yaowan.png">
  <link rel="mask-icon" href="/images/logo-yaowan.svg" color="#49b1f5">

<link rel="stylesheet" href="/css/main.css">


<link rel="stylesheet" href="/lib/font-awesome/css/font-awesome.min.css">
  <link rel="stylesheet" href="//cdn.jsdelivr.net/gh/fancyapps/fancybox@3/dist/jquery.fancybox.min.css">
  <link rel="stylesheet" href="//cdn.jsdelivr.net/npm/pace-js@1/themes/blue/pace-theme-minimal.css">
  <script src="//cdn.jsdelivr.net/npm/pace-js@1/pace.min.js"></script>

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"zhouyuqian.com","root":"/","scheme":"Mist","version":"7.7.2","exturl":false,"sidebar":{"position":"right","display":"post","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":true,"show_result":"default","style":"flat"},"back2top":{"enable":true,"sidebar":false,"scrollpercent":true},"bookmark":{"enable":true,"color":"#49b1f5","save":"auto"},"fancybox":true,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":"valine","storage":true,"lazyload":false,"nav":null},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":"ture"},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},"path":"search.xml"};
  </script>

  <meta name="description" content="gm&#x2F;id 设计方法的优点  gm&#x2F;Id对应Vov，通过其数值大小的选取来达到增益与带宽的折衷； gm&#x2F;Id方法是一种look-up table方法； gm&#x2F;Id方法为短沟道器件电路设计提供了比公式手算更准确的初值； gm&#x2F;Id方法为亚阈值设计提供了有力的工具。">
<meta property="og:type" content="article">
<meta property="og:title" content="gmid 设计方法">
<meta property="og:url" content="https://zhouyuqian.com/2021/03/10/gmid/index.html">
<meta property="og:site_name" content="Fitz&#39;s Blog - 快乐学习每一天">
<meta property="og:description" content="gm&#x2F;id 设计方法的优点  gm&#x2F;Id对应Vov，通过其数值大小的选取来达到增益与带宽的折衷； gm&#x2F;Id方法是一种look-up table方法； gm&#x2F;Id方法为短沟道器件电路设计提供了比公式手算更准确的初值； gm&#x2F;Id方法为亚阈值设计提供了有力的工具。">
<meta property="og:locale" content="zh_CN">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727181920.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727181933.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727181942.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182000.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182013.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182032.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182044.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182058.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182116.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182131.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182143.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182154.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182204.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182215.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727181843.png">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182230.svg">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182245.svg">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182256.svg">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727182322.svg">
<meta property="og:image" content="https://pic.zhouyuqian.com/img/20210727181844.svg">
<meta property="article:published_time" content="2021-03-10T02:34:12.000Z">
<meta property="article:modified_time" content="2021-03-16T02:34:12.000Z">
<meta property="article:author" content="Fitz">
<meta property="article:tag" content="IC_design">
<meta property="article:tag" content="Analog">
<meta property="article:tag" content="gmid">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://pic.zhouyuqian.com/img/20210727181920.png">

<link rel="canonical" href="https://zhouyuqian.com/2021/03/10/gmid/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true
  };
</script>

  <title>gmid 设计方法 | Fitz's Blog - 快乐学习每一天</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript><!-- hexo-inject:begin --><!-- hexo-inject:end -->

</head>

<body itemscope itemtype="http://schema.org/WebPage">
  <!-- hexo-inject:begin --><!-- hexo-inject:end --><div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <div>
      <a href="/" class="brand" rel="start">
        <span class="logo-line-before"><i></i></span>
        <span class="site-title">Fitz's Blog - 快乐学习每一天</span>
        <span class="logo-line-after"><i></i></span>
      </a>
    </div>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>


<nav class="site-nav">
  
  <ul id="menu" class="menu">
        <li class="menu-item menu-item-home">

    <a href="/" rel="section"><i class="fa fa-fw fa-home"></i>首页</a>

  </li>
        <li class="menu-item menu-item-about">

    <a href="/about/" rel="section"><i class="fa fa-fw fa-user"></i>关于</a>

  </li>
        <li class="menu-item menu-item-tags">

    <a href="/tags/" rel="section"><i class="fa fa-fw fa-tags"></i>标签</a>

  </li>
        <li class="menu-item menu-item-categories">

    <a href="/categories/" rel="section"><i class="fa fa-fw fa-th"></i>分类</a>

  </li>
        <li class="menu-item menu-item-archives">

    <a href="/archives/" rel="section"><i class="fa fa-fw fa-archive"></i>归档</a>

  </li>
        <li class="menu-item menu-item-ariang">

    <a href="/AriaNg/" rel="section"><i class="fa fa-fw fa-cloud-download"></i>AriaNg</a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>

</nav>
  <div class="site-search">
    <div class="popup search-popup">
    <div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container">
    <input autocomplete="off" autocorrect="off" autocapitalize="off"
           placeholder="搜索..." spellcheck="false"
           type="search" class="search-input">
  </div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div id="search-result"></div>

</div>
<div class="search-pop-overlay"></div>

  </div>
</div>
    </header>

    
  <div class="back-to-top">
    <i class="fa fa-arrow-up"></i>
    <span>0%</span>
  </div>
  <div class="reading-progress-bar"></div>
  <a role="button" class="book-mark-link book-mark-link-fixed"></a>

  <a href="https://github.com/Starrynightzyq" class="github-corner" title="Follow me on GitHub" aria-label="Follow me on GitHub" rel="noopener" target="_blank"><svg width="80" height="80" viewBox="0 0 250 250" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content">
            

  <div class="posts-expand">
      
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block " lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="https://zhouyuqian.com/2021/03/10/gmid/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/images/avatar.jpg">
      <meta itemprop="name" content="Fitz">
      <meta itemprop="description" content="">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="Fitz's Blog - 快乐学习每一天">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          gmid 设计方法
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="fa fa-calendar-o"></i>
              </span>
              <span class="post-meta-item-text">发表于</span>

              <time title="创建时间：2021-03-10 10:34:12" itemprop="dateCreated datePublished" datetime="2021-03-10T10:34:12+08:00">2021-03-10</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="fa fa-calendar-check-o"></i>
                </span>
                <span class="post-meta-item-text">更新于</span>
                <time title="修改时间：2021-03-16 10:34:12" itemprop="dateModified" datetime="2021-03-16T10:34:12+08:00">2021-03-16</time>
              </span>
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="fa fa-folder-o"></i>
              </span>
              <span class="post-meta-item-text">分类于</span>
                <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
                  <a href="/categories/Analog/" itemprop="url" rel="index"><span itemprop="name">Analog</span></a>
                </span>
            </span>

          <br>
            <span class="post-meta-item" title="本文字数">
              <span class="post-meta-item-icon">
                <i class="fa fa-file-word-o"></i>
              </span>
                <span class="post-meta-item-text">本文字数：</span>
              <span>4k</span>
            </span>
            <span class="post-meta-item" title="阅读时长">
              <span class="post-meta-item-icon">
                <i class="fa fa-clock-o"></i>
              </span>
                <span class="post-meta-item-text">阅读时长 &asymp;</span>
              <span>4 分钟</span>
            </span>

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <h1 id="gmid-设计方法的优点">gm/id 设计方法的优点</h1>
<ul>
<li>gm/Id对应Vov，通过其数值大小的选取来达到增益与带宽的折衷；</li>
<li>gm/Id方法是一种look-up table方法；</li>
<li>gm/Id方法为短沟道器件电路设计提供了比公式手算更准确的初值；</li>
<li>gm/Id方法为亚阈值设计提供了有力的工具。</li>
</ul>
<a id="more"></a>
<h1 id="长沟道模型回顾">长沟道模型回顾</h1>
<p>在长沟道期间中，<span class="math inline">\(V_{ov}\)</span> 表示过驱动电压： <span class="math display">\[
V_{ov} = V_{gs} - V_{T}
\]</span></p>
<p><img src="https://pic.zhouyuqian.com/img/20210727181920.png" alt="image-20210310201310417" style="zoom:50%;"></p>
<p>在不同的 <span class="math inline">\(V_{ds}\)</span> 和 <span class="math inline">\(V_{gs}\)</span> 下，NMOS 管会工作在不同的区域：</p>
<ul>
<li><p>截止区：</p>
<p><span class="math inline">\(V_{ov}&lt;0\)</span> ( or <span class="math inline">\(V_{gs} &lt; V_{T}\)</span> )，</p>
<p><span class="math display">\[
I_{D} = 0
\]</span></p></li>
<li><p>线性区：</p>
<p><span class="math inline">\(V_{ov}&gt;0\)</span> and <span class="math inline">\(V_{ds} &lt; V_{dsat}\)</span>，</p>
<p><span class="math display">\[
I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L}[2V_{ov}V_{DS}-V_{DS}^2]
\]</span></p>
<p><span class="math display">\[
R_{on} = \frac{\partial V_{ds}}{\partial I_D}
\]</span></p></li>
<li><p>饱和区：</p>
<p><span class="math inline">\(V_{ov}&gt;0\)</span> and <span class="math inline">\(V_{ds} &gt; V_{dsat}\)</span>，</p>
<p><span class="math display">\[
I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L} V_{ov}^2
\]</span></p></li>
</ul>
<p><img src="https://pic.zhouyuqian.com/img/20210727181933.png" alt="image-20210310202752762" style="zoom:50%;"></p>
<p><img src="https://pic.zhouyuqian.com/img/20210727181942.png" alt="image-20210310202909324" style="zoom:50%;"></p>
<p>在<strong>饱和区</strong>，<span class="math inline">\(I_D\)</span> 是 <span class="math inline">\(V_{ov}\)</span> 的函数，小信号模型中的 <span class="math inline">\(g_m\)</span> 被定义为： <span class="math display">\[
g_m = \frac{\partial I_D}{\partial V_{ov}} = \mu C_{ox}\frac{W}{L} V_{ov}
\]</span> 可以得到： <span class="math display">\[
\frac{g_m}{I_D} = \frac{2}{V_{ov}}
\]</span> 上述公式表明了 <span class="math inline">\(g_m/I_D\)</span> 和 <span class="math inline">\(V_{ov}\)</span> 之间的关系。</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182000.png" alt="image-20210310213132752" style="zoom:50%;"></p>
<p>定义晶体管的截止频率 (transit frequency) <span class="math inline">\(f_T\)</span>: <span class="math display">\[
f_T = \frac{1}{2\pi}\frac{g_m}{C_{gs}}
\]</span> 其中 <span class="math inline">\(C_{gs} = \frac{2}{3}C_{ox}WL\)</span>，则： <span class="math display">\[
f_T = \frac{1}{2\pi}\frac{g_m}{C_{gs}} = \frac{1}{2\pi}\frac{2\mu V_{ov}}{3L^2}
\]</span> 从上面的等式中我们可以得到 <span class="math inline">\(g_m\)</span> 和 <span class="math inline">\(f_T\)</span> 之间的约束，Fig. 9 将其表示出来。</p>
<h1 id="传统的-v_ov-设计方法">传统的 <span class="math inline">\(V_{ov}\)</span> 设计方法</h1>
<p><img src="https://pic.zhouyuqian.com/img/20210727182013.png" alt="image-20210310203850046" style="zoom:50%;"></p>
<p>在上图的电路中，假设 <span class="math inline">\(V_{ov} = 300 mV, I_D=1mA\)</span>，则： <span class="math display">\[
v_{out} = -i_s \cdot R_L=-v_{in} \cdot g_m \cdot 1k\Omega
\]</span></p>
<p><span class="math display">\[
g_m = 2\frac{I_D}{V_{ov}} = \frac{2\cdot 1mA}{300 mV} = 6.7 \frac{mA}{V}
\]</span></p>
<p>可以得到增益为： <span class="math display">\[
\frac{v_{out}}{v_{in}}= -g_m \cdot R_L = -6.7\frac{mA}{V}\cdot 1k\Omega = -6.7 \frac{V}{V}
\]</span> 在上面的结构中假设要求带宽 <span class="math inline">\(500MHz\)</span>，增益为 10，基于 <span class="math inline">\(V_{ov}\)</span> 的设计流程如下：</p>
<ol type="1">
<li><p>为了保证增益，首先可以计算 <span class="math inline">\(g_m\)</span>： <span class="math display">\[
g_m = \frac{v_{out}}{v_{in}} / R_L = 10\frac{V}{V}/1k = 10mA/V
\]</span></p></li>
<li><p>输入极点在 <span class="math inline">\(500MHz\)</span>，可以以此计算 <span class="math inline">\(C_{gs}\)</span>： <span class="math display">\[
C_{gs} = \frac{1}{2\pi \times 300\Omega \times 500MHz} = 1.1pF
\]</span></p></li>
<li><p>可以计算出截止频率： <span class="math display">\[
f_T = \frac{g_m}{C_{gs}}=\frac{10mS}{1.1pF} = 9.4 GHz
\]</span></p></li>
<li><p>通过 Fig. 9 可以知道： <span class="math display">\[
V_{ov} \geq 75 mV
\]</span></p></li>
<li><p>由此通过 Fig. 9 可以知道： <span class="math display">\[
g_m/I_D \leq 26 mS/mA
\]</span></p></li>
<li><p>最终可以得到： <span class="math display">\[
I_D = \frac{g_m}{g_m/I_D} = \frac{10mS}{26mS/mA}=385\mu A
\]</span></p></li>
</ol>
<h1 id="基于-v_ov-设计方法的缺点">基于 <span class="math inline">\(V_{ov}\)</span> 设计方法的缺点</h1>
<p><img src="https://pic.zhouyuqian.com/img/20210727182032.png" alt="image-20210310220510541" style="zoom:50%;"></p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182044.png" alt="image-20210310220527243" style="zoom:50%;"></p>
<p>从 Fig.12 可以看出，长沟道模型在 <span class="math inline">\(V_{ov}\)</span> 较小 (弱反型区) 时 <span class="math inline">\(g_m/I_D\)</span> 与 <span class="math inline">\(V_{ov}\)</span> 的关系预测值与仿真值相差较大，在亚阈值区 (<span class="math inline">\(V_{ov} &lt;0\)</span>) 则完全失效。</p>
<p>从 Fig.12 可以看出，长沟道模型预测的 <span class="math inline">\(f_T\)</span> 与 <span class="math inline">\(V_{ov}\)</span> 的关系与仿真值相差较大。</p>
<h1 id="基于-g_mi_d-的设计方法">基于 <span class="math inline">\(g_m/I_D\)</span> 的设计方法</h1>
<p>由于 <span class="math inline">\(g_m/I_D\)</span> 与 <span class="math inline">\(V_{ov}\)</span> 之间存在关系，则可以用 <span class="math inline">\(g_m/I_D\)</span> 取代 <span class="math inline">\(V_{ov}\)</span> 来表示与 <span class="math inline">\(f_T\)</span> 的关系，原来 Fig.9 的关系可以用 Fig.14 来表示。</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182058.png" alt="image-20210310221848963" style="zoom:50%;"></p>
<h2 id="设计流程">设计流程</h2>
<p>在 Fig.8 的结构中假设要求带宽 <span class="math inline">\(500MHz\)</span>，增益为 10，基于 <span class="math inline">\(g_m/I_D\)</span> 的设计流程如下：</p>
<ol type="1">
<li><p>为了保证增益，首先可以计算 <span class="math inline">\(g_m\)</span>： <span class="math display">\[
g_m = \frac{v_{out}}{v_{in}} / R_L = 10\frac{V}{V}/1k = 10mA/V
\]</span></p></li>
<li><p>输入极点在 <span class="math inline">\(500MHz\)</span>，可以以此计算 <span class="math inline">\(C_{gs}\)</span>： <span class="math display">\[
C_{gs} = \frac{1}{2\pi \times 300\Omega \times 500MHz} = 1.1pF
\]</span></p></li>
<li><p>可以计算出截止频率： <span class="math display">\[
f_T = \frac{g_m}{C_{gs}}=\frac{10mS}{1.1pF} = 9.4 GHz
\]</span></p></li>
<li><p>通过 Fig.14 可以知道： <span class="math display">\[
g_m/I_D \leq 17.5 mS/mA
\]</span></p></li>
<li><p>最终： <span class="math display">\[
I_D = \frac{g_m}{g_m/I_D} = \frac{10mS}{17.5mS/mA}=570\mu A
\]</span></p></li>
</ol>
<p><img src="https://pic.zhouyuqian.com/img/20210727182116.png" alt="image-20210311101408375" style="zoom:50%;"></p>
<p>Fig.16 是不同的 <span class="math inline">\(L\)</span> 下，<span class="math inline">\(f_T\)</span> 与 <span class="math inline">\(g_m/I_D\)</span> 的关系，<strong>越大的 <span class="math inline">\(L\)</span> 通常意味着晶体管更慢</strong>。这就意味着要是没有其他的限制，我们就会选择最小的 <span class="math inline">\(L\)</span>，这样晶体管速度最快，面积也最小。因此还需要一个限制会影响 <span class="math inline">\(L\)</span> 的选取，这个限制就是 <span class="math inline">\(r_0\)</span>。</p>
<h2 id="考虑-r_0">考虑 <span class="math inline">\(r_0\)</span></h2>
<p><img src="https://pic.zhouyuqian.com/img/20210727182131.png" alt="image-20210311095357398" style="zoom:50%;"></p>
<p>从上图可以看到 <span class="math inline">\(r_0\)</span> 相当于一个和 <span class="math inline">\(R_L\)</span> 并联的负载，但我们考虑 $R_L $ 时，可以忽略 <span class="math inline">\(R_L\)</span> 只考虑 <span class="math inline">\(r_0\)</span>。则： <span class="math display">\[
Intrinsic \ \ Gain = g_mr_0 (\frac{V}{V})
\]</span> <strong><span class="math inline">\(Intrinsic \ \ Gain\)</span> 是晶体管可以达到的最大增益</strong>。</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182143.png" alt="image-20210311100239741" style="zoom:50%;"></p>
<p>从 Fig.19 中可以看到不同的 <span class="math inline">\(L\)</span> 对应不同的 <span class="math inline">\(g_mr_0(Intrinsic \ \ Gain)\)</span>。</p>
<p>由于 <span class="math inline">\(r_0\)</span> 与 <span class="math inline">\(V_{ds}\)</span> 有关，这里的假设是 <span class="math inline">\(V_{ds} = V_{DD}/2\)</span>。</p>
<h2 id="考虑-w">考虑 <span class="math inline">\(W\)</span></h2>
<p>通过 Fig.16 和 Fig.19 可以确定 <span class="math inline">\(L\)</span> 和 <span class="math inline">\(g_m/I_D\)</span>：为了满足增益的要求，通过 Fig.19 确定了 <span class="math inline">\(L\)</span>；为了满足 <span class="math inline">\(f_T\)</span> 的要求，通过 Fig.16 确定了 <span class="math inline">\(g_m/I_D\)</span>；通过 <span class="math inline">\(g_m/I_D\)</span> 和 <span class="math inline">\(g_m\)</span> 确定了 <span class="math inline">\(I_D\)</span>。最后剩下的没有确定的量就是 <span class="math inline">\(W\)</span>。</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182154.png" alt="image-20210311102457544" style="zoom:50%;"></p>
<p><img src="https://pic.zhouyuqian.com/img/20210727182204.png" alt="image-20210311104821336" style="zoom:50%;"></p>
<p>在 Fig.21 中，由于 <span class="math inline">\(V_b\)</span> 和 <span class="math inline">\(V_{DS}\)</span> 相等，因此 a 和 b 的 <span class="math inline">\(g_m\)</span> 、<span class="math inline">\(C_{gs}\)</span> 相同，而 b 由于并联了两个晶体管，由此 <span class="math inline">\(I_D\)</span> 是 a 的两倍，c 和 b 等价。由此只要保证 <span class="math inline">\(g_m\)</span>-to-<span class="math inline">\(I_D\)</span> 和 <span class="math inline">\(g_m\)</span>-to-<span class="math inline">\(C_{gs}\)</span> 的比率相等，W 变为原来的 N 倍，<span class="math inline">\(I_D\)</span> 也变为原来的 N 倍，而 <span class="math inline">\(g_m/I_D\)</span> 和 <span class="math inline">\(f_T\)</span> 不变。</p>
<h1 id="a-top-to-bottom-design-example">A Top-to-Bottom Design Example</h1>
<p><img src="https://pic.zhouyuqian.com/img/20210727182215.png" alt="image-20210311095035057" style="zoom:50%;"></p>
<p>设计一个差分放大器，结构如 Fig.23 所示，要求：</p>
<ul>
<li>Gain of ≈ 10</li>
<li>Bandwidth of ≈ 200MHz</li>
<li>Drive a 1pF load</li>
<li>Be driven by a 300Ω</li>
<li>Lowest possible power</li>
</ul>
<h2 id="搭建电路">搭建电路</h2>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727181843.png" alt="OPA"><figcaption aria-hidden="true">OPA</figcaption>
</figure>
<h2 id="获取晶体管参数">获取晶体管参数</h2>
<p>通过 cadence 仿真并用 MATLAB 处理数据并画出图形。</p>
<center>
<img style="border-radius: 0.3125em;    box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="https://pic.zhouyuqian.com/img/20210727182230.svg"><br>
<br><br>

<div style="color:orange; border-bottom: 1px solid #d9d9d9;    display: inline-block;    color: #999;    padding: 2px;">
Figure 25: Fruquency and Gain Tradeoff
</div>
</center>
<center>
<img style="border-radius: 0.3125em;    box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="https://pic.zhouyuqian.com/img/20210727182245.svg"><br>
<br><br>

<div style="color:orange; border-bottom: 1px solid #d9d9d9;    display: inline-block;    color: #999;    padding: 2px;">
Figure 26
</div>
</center>
<center>
<img style="border-radius: 0.3125em;    box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="https://pic.zhouyuqian.com/img/20210727182256.svg"><br>
<br><br>

<div style="color:orange; border-bottom: 1px solid #d9d9d9;    display: inline-block;    color: #999;    padding: 2px;">
Figure 27
</div>
</center>
<center>
<img style="border-radius: 0.3125em;    box-shadow: 0 2px 4px 0 rgba(34,36,38,.12),0 2px 10px 0 rgba(34,36,38,.08);" src="https://pic.zhouyuqian.com/img/20210727182322.svg"><br>
<br><br>

<div style="color:orange; border-bottom: 1px solid #d9d9d9;    display: inline-block;    color: #999;    padding: 2px;">
Figure 28
</div>
</center>
<h2 id="选择参数">选择参数</h2>
<ol type="1">
<li><p>首先确定 <span class="math inline">\(L\)</span>，通过 Fig.25，选取 <span class="math inline">\(L=340n m\)</span>，在 <span class="math inline">\(g_m\)</span> 选取中等值 (15) 时，Intrinsic Gain 大约是 100，这意味着它只会对整体增益产生10％的影响。 <span class="math display">\[
L = 340n
\]</span></p></li>
<li><p>下一步是计算 R 的值： <span class="math display">\[
R = \frac{1}{2\pi \times C_L \times 200MHz} = 800 \Omega
\]</span></p></li>
<li><p>因此通过增益可以确定 <span class="math inline">\(g_m\)</span>： <span class="math display">\[
g_m = \frac{Gain}{R}= 12.5mS
\]</span></p></li>
<li><p>由于 200MHz 并不是唯一的一个极点，300Ω 的输入电阻与 <span class="math inline">\(C_{gs}\)</span> 会产生第二个极点，为了减少第二个极点对系统的影响，将第二个极点设置在第一个极点的10倍处： <span class="math display">\[
C_{gs} = \frac{1}{2\pi \times 300\Omega \times 2GHz} = 265 fF
\]</span></p></li>
<li><p>有了 <span class="math inline">\(g_m\)</span> 和 <span class="math inline">\(C_{gs}\)</span>，可以计算出截止频率： <span class="math display">\[
f_{T} = \frac{1}{2\pi}\frac{g_m}{C_{gs}} = \frac{1}{2\pi}\frac{12.5mS}{265fF}=7.5GHz
\]</span></p></li>
<li><p>有了 <span class="math inline">\(L\)</span> 和 <span class="math inline">\(f_T\)</span>，通过 Fig.25 可以得到 <span class="math inline">\(g_m/I_D\)</span>： <span class="math display">\[
g_m/I_D = 15mS/mA
\]</span></p></li>
<li><p>有了 <span class="math inline">\(g_m/I_D\)</span> 和 <span class="math inline">\(g_m\)</span>，可以计算出 <span class="math inline">\(I_D\)</span>： <span class="math display">\[
I_D = \frac{g_m}{g_m/I_D}= \frac{12.5}{15}\approx 0.83mA
\]</span></p></li>
<li><p>最后，通过 Fig.27 找出对应的 <span class="math inline">\(I_D/(W/L)\)</span> 为 <span class="math inline">\(2.2046e-06\)</span>，因此可以计算出 <span class="math inline">\(W/L = 376\)</span>： <span class="math display">\[
W = 128um
\]</span></p></li>
</ol>
<h2 id="仿真结果">仿真结果</h2>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727181844.svg" alt="db"><figcaption aria-hidden="true">db</figcaption>
</figure>
<p>仿真的结果显示增益为 18.7 dB，带宽为 164.4 MHz。</p>
<h1 id="reference">Reference</h1>
<p>[1] <a target="_blank" rel="noopener" href="http://pan.baidu.com/s/1slW1U4P">Stanford ee214b的课件</a></p>

    </div>

    
    
    

    <div>
      
        <div>
    
        <div style="text-align:center;color: #ccc;font-size:14px;">-------The end of this article&nbsp;<i class="fa fa-paw"></i>&nbsp;Thank you for your reading-------</div>
    
</div>
      
    </div>
        

<div>
<ul class="post-copyright">
  <li class="post-copyright-author">
    <strong>本文作者： </strong>Fitz
  </li>
  <li class="post-copyright-link">
    <strong>本文链接：</strong>
    <a href="https://zhouyuqian.com/2021/03/10/gmid/" title="gmid 设计方法">https://zhouyuqian.com/2021/03/10/gmid/</a>
  </li>
  <li class="post-copyright-license">
    <strong>版权声明： </strong>本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/zh" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i>BY-NC-SA</a> 许可协议。转载请注明出处！
  </li>
</ul>
</div>


      <footer class="post-footer">
          <div class="post-tags">
              <a href="/tags/IC-design/" <i class="fa fa-tag"></i> IC_design</a>
              <a href="/tags/Analog/" <i class="fa fa-tag"></i> Analog</a>
              <a href="/tags/gmid/" <i class="fa fa-tag"></i> gmid</a>
          </div>

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/2021/03/07/VirtuosoOnUbuntu/" rel="prev" title="Cadence IC617 on Ubuntu">
      <i class="fa fa-chevron-left"></i> Cadence IC617 on Ubuntu
    </a></div>
      <div class="post-nav-item">
    <a href="/2021/03/15/VerilogOnVscode/" rel="next" title="配置 VS Code 的 Verilog 开发环境">
      配置 VS Code 的 Verilog 开发环境 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  

  </div>


          </div>
          

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#gmid-%E8%AE%BE%E8%AE%A1%E6%96%B9%E6%B3%95%E7%9A%84%E4%BC%98%E7%82%B9"><span class="nav-number">1.</span> <span class="nav-text">gm&#x2F;id 设计方法的优点</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#%E9%95%BF%E6%B2%9F%E9%81%93%E6%A8%A1%E5%9E%8B%E5%9B%9E%E9%A1%BE"><span class="nav-number">2.</span> <span class="nav-text">长沟道模型回顾</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#%E4%BC%A0%E7%BB%9F%E7%9A%84-v_ov-%E8%AE%BE%E8%AE%A1%E6%96%B9%E6%B3%95"><span class="nav-number">3.</span> <span class="nav-text">传统的 \(V_{ov}\) 设计方法</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#%E5%9F%BA%E4%BA%8E-v_ov-%E8%AE%BE%E8%AE%A1%E6%96%B9%E6%B3%95%E7%9A%84%E7%BC%BA%E7%82%B9"><span class="nav-number">4.</span> <span class="nav-text">基于 \(V_{ov}\) 设计方法的缺点</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#%E5%9F%BA%E4%BA%8E-g_mi_d-%E7%9A%84%E8%AE%BE%E8%AE%A1%E6%96%B9%E6%B3%95"><span class="nav-number">5.</span> <span class="nav-text">基于 \(g_m&#x2F;I_D\) 的设计方法</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%AE%BE%E8%AE%A1%E6%B5%81%E7%A8%8B"><span class="nav-number">5.1.</span> <span class="nav-text">设计流程</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%80%83%E8%99%91-r_0"><span class="nav-number">5.2.</span> <span class="nav-text">考虑 \(r_0\)</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%80%83%E8%99%91-w"><span class="nav-number">5.3.</span> <span class="nav-text">考虑 \(W\)</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#a-top-to-bottom-design-example"><span class="nav-number">6.</span> <span class="nav-text">A Top-to-Bottom Design Example</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#%E6%90%AD%E5%BB%BA%E7%94%B5%E8%B7%AF"><span class="nav-number">6.1.</span> <span class="nav-text">搭建电路</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E8%8E%B7%E5%8F%96%E6%99%B6%E4%BD%93%E7%AE%A1%E5%8F%82%E6%95%B0"><span class="nav-number">6.2.</span> <span class="nav-text">获取晶体管参数</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E9%80%89%E6%8B%A9%E5%8F%82%E6%95%B0"><span class="nav-number">6.3.</span> <span class="nav-text">选择参数</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#%E4%BB%BF%E7%9C%9F%E7%BB%93%E6%9E%9C"><span class="nav-number">6.4.</span> <span class="nav-text">仿真结果</span></a></li></ol></li><li class="nav-item nav-level-1"><a class="nav-link" href="#reference"><span class="nav-number">7.</span> <span class="nav-text">Reference</span></a></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="Fitz"
      src="/images/avatar.jpg">
  <p class="site-author-name" itemprop="name">Fitz</p>
  <div class="site-description" itemprop="description"></div>
</div>
<div class="site-state-wrap motion-element">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/archives/">
        
          <span class="site-state-item-count">78</span>
          <span class="site-state-item-name">日志</span>
        </a>
      </div>
      <div class="site-state-item site-state-categories">
            <a href="/categories/">
          
        <span class="site-state-item-count">25</span>
        <span class="site-state-item-name">分类</span></a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/tags/">
          
        <span class="site-state-item-count">52</span>
        <span class="site-state-item-name">标签</span></a>
      </div>
  </nav>
</div>
  <div class="links-of-author motion-element">
      <span class="links-of-author-item">
        <a href="https://github.com/Starrynightzyq" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;Starrynightzyq" rel="noopener" target="_blank"><i class="fa fa-fw fa-github"></i>GitHub</a>
      </span>
      <span class="links-of-author-item">
        <a href="mailto:starrynightzyq@gmail.com" title="E-Mail → mailto:starrynightzyq@gmail.com" rel="noopener" target="_blank"><i class="fa fa-fw fa-envelope"></i>E-Mail</a>
      </span>
      <span class="links-of-author-item">
        <a href="https://space.bilibili.com/57711842" title="Bilibili → https:&#x2F;&#x2F;space.bilibili.com&#x2F;57711842" rel="noopener" target="_blank"><i class="fa fa-fw fa-custom bilibili"></i>Bilibili</a>
      </span>
      <span class="links-of-author-item">
        <a href="http://wpa.qq.com/msgrd?v=3&uin=2421854978&site=qq&menu=yes" title="QQ → http:&#x2F;&#x2F;wpa.qq.com&#x2F;msgrd?v&#x3D;3&amp;uin&#x3D;2421854978&amp;site&#x3D;qq&amp;menu&#x3D;yes" rel="noopener" target="_blank"><i class="fa fa-fw fa-custom qq"></i>QQ</a>
      </span>
  </div>


  <div class="links-of-blogroll motion-element">
    <div class="links-of-blogroll-title">
      <i class="fa fa-fw fa-link"></i>
      Links
    </div>
    <ul class="links-of-blogroll-list">
        <li class="links-of-blogroll-item">
          <a href="http://leiblog.wang/" title="http:&#x2F;&#x2F;leiblog.wang" rel="noopener" target="_blank">Lei Blog</a>
        </li>
    </ul>
  </div>

      </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        
  <div class="beian"><a href="http://beian.miit.gov.cn/" rel="noopener" target="_blank">苏ICP备20042287号 </a>
  </div>

<div class="copyright">
  
  &copy; 2018 – 
  <span itemprop="copyrightYear">2024</span>
  <span class="with-love">
    <i class="fa fa-user"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">Fitz</span>
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item-icon">
      <i class="fa fa-area-chart"></i>
    </span>
    <span title="站点总字数">251k</span>
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item-icon">
      <i class="fa fa-coffee"></i>
    </span>
    <span title="站点阅读时长">3:48</span>
</div>
  <div class="powered-by">由 <a href="https://hexo.io/" class="theme-link" rel="noopener" target="_blank">Hexo</a> 强力驱动 v5.3.0
  </div>
  <span class="post-meta-divider">|</span>
  <div class="theme-info">主题 – <a href="https://mist.theme-next.org/" class="theme-link" rel="noopener" target="_blank">NexT.Mist</a> v7.7.2
  </div>

        








      </div>
    </footer>
  </div>

  
  <script src="/lib/anime.min.js"></script>
  <script src="//cdn.jsdelivr.net/gh/theme-next/theme-next-pjax@0/pjax.min.js"></script>
  <script src="//cdn.jsdelivr.net/npm/jquery@3/dist/jquery.min.js"></script>
  <script src="//cdn.jsdelivr.net/gh/fancyapps/fancybox@3/dist/jquery.fancybox.min.js"></script>
  <script src="/lib/velocity/velocity.min.js"></script>
  <script src="/lib/velocity/velocity.ui.min.js"></script>

<script src="/js/utils.js"></script>

<script src="/js/motion.js"></script>


<script src="/js/schemes/muse.js"></script>


<script src="/js/next-boot.js"></script>

<script src="/js/bookmark.js"></script>

  <script>
var pjax = new Pjax({
  selectors: [
    'head title',
    '#page-configurations',
    '.content-wrap',
    '.post-toc-wrap',
    '#pjax'
  ],
  switches: {
    '.post-toc-wrap': Pjax.switches.innerHTML
  },
  analytics: false,
  cacheBust: false,
  scrollTo : !CONFIG.bookmark.enable
});

window.addEventListener('pjax:success', () => {
  document.querySelectorAll('script[pjax], script#page-configurations, #pjax script').forEach(element => {
    var code = element.text || element.textContent || element.innerHTML || '';
    var parent = element.parentNode;
    parent.removeChild(element);
    var script = document.createElement('script');
    if (element.id) {
      script.id = element.id;
    }
    if (element.className) {
      script.className = element.className;
    }
    if (element.type) {
      script.type = element.type;
    }
    if (element.src) {
      script.src = element.src;
      // Force synchronous loading of peripheral JS.
      script.async = false;
    }
    if (element.getAttribute('pjax') !== null) {
      script.setAttribute('pjax', '');
    }
    if (code !== '') {
      script.appendChild(document.createTextNode(code));
    }
    parent.appendChild(script);
  });
  NexT.boot.refresh();
  // Define Motion Sequence & Bootstrap Motion.
  if (CONFIG.motion.enable) {
    NexT.motion.integrator
      .init()
      .add(NexT.motion.middleWares.subMenu)
      .add(NexT.motion.middleWares.postList)
      .bootstrap();
  }
  NexT.utils.updateSidebarPosition();
});
</script>




  




  
<script src="/js/local-search.js"></script>













    <div id="pjax">
  

  
      

<script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      loader: {
        source: {
          '[tex]/amsCd': '[tex]/amscd',
          '[tex]/AMScd': '[tex]/amscd'
        }
      },
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              let target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    (function () {
      var script = document.createElement('script');
      script.src = '//cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js';
      script.defer = true;
      document.head.appendChild(script);
    })();
  } else {
    MathJax.startup.document.state(0);
    MathJax.texReset();
    MathJax.typeset();
  }
</script>

    

  

    </div>

  <!-- 页面点击小红心 -->
  <!-- <script type="text/javascript" src="/js/jquery-3.3.1.min.js"></script> -->
  <script type="text/javascript" src="/js/clicklove.js"></script><!-- hexo-inject:begin --><!-- hexo-inject:end -->
</body>
</html>
